Author 
Comment/Response 
Bill Simpson

01/07/13 5:12pm
Since you have an explicit equation for p[x] in your system and you only use p'[x] in a single place, you might have more success if you differentiate your p[x] equation, substitute that for your p'[x], eliminate your initial condition for p[10] by determining what that really defines for the rest of your system and then try to use NDSolve for {a,u}.
p[x] == 2 (1  a[x]^(3/2))  50 a''[x] + 1
so
p'[x] == 3*a'[x]/a[x]^(5/2)  50*a'''[x]
and substitute that for p'[x].
Unfortunately I can't see how to then incorporate the p[10] into the result, but perhaps you can.
Sometimes Solve and NDSolve have fewer difficulties if you solve for all the variables instead of just solving for a subset. That may apply in your case.
I am also concerned that you have a[x]^(3/2). It is possible that this is the reason for the errors that you are getting. If you can justify rearranging the equations of your system to get that a[x] out of the denominator then this might also help find a solution.
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